Article An Alternative to Dark Matter and Dark Energy: Scale-Dependent Gravity in Superfluid Vacuum Theory
Konstantin G. Zloshchastiev
Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa; [email protected]
Received: 29 August 2020; Accepted: 10 October 2020; Published: 15 October 2020
Abstract: We derive an effective gravitational potential, induced by the quantum wavefunction of a physical vacuum of a self-gravitating configuration, while the vacuum itself is viewed as the superfluid described by the logarithmic quantum wave equation. We determine that gravity has a multiple-scale pattern, to such an extent that one can distinguish sub-Newtonian, Newtonian, galactic, extragalactic and cosmological terms. The last of these dominates at the largest length scale of the model, where superfluid vacuum induces an asymptotically Friedmann–Lemaître–Robertson–Walker-type spacetime, which provides an explanation for the accelerating expansion of the Universe. The model describes different types of expansion mechanisms, which could explain the discrepancy between measurements of the Hubble constant using different methods. On a galactic scale, our model explains the non-Keplerian behaviour of galactic rotation curves, and also why their profiles can vary depending on the galaxy. It also makes a number of predictions about the behaviour of gravity at larger galactic and extragalactic scales. We demonstrate how the behaviour of rotation curves varies with distance from a gravitating center, growing from an inner galactic scale towards a metagalactic scale: A squared orbital velocity’s profile crosses over from Keplerian to flat, and then to non-flat. The asymptotic non-flat regime is thus expected to be seen in the outer regions of large spiral galaxies.
Keywords: quantum gravity; cosmology; superfluid vacuum; emergent spacetime; dark matter; galactic rotation curve; quantum Bose liquid; logarithmic fluid; logarithmic wave equation
1. Introduction Astronomical observations over many length scales support the existence of a number of novel phenomena, which are usually attributed to dark matter (DM) and dark energy (DE). Dark matter was introduced to explain a range of observed phenomena at a galactic scale, such as flat rotation curves, while dark energy is expected to account for cosmological-scale dynamics, such as the accelerating expansion of the Universe. For instance, the ΛCDM model, which is currently the most popular approach used in cosmology and galaxy-scale astrophysics, makes use of both DE and cold DM concepts [1]. In spite of being a generally successful framework purporting to explain the large-scale structure of the Universe, it currently faces certain challenges [2,3]. There is also growing consensus that a convincing theory of DM- and DE-attributed phenomena cannot be a stand-alone model; but should, instead, be a part of a fundamental theory involving all known interactions. In turn, we contend that formulating this fundamental theory will be impossible without a clear understanding of the dynamical structure of the physical vacuum, which underlies all interactions that we know of. Moreover, this theory must operate at a quantum level, which necessitates us rethinking of the concept of gravity using basic notions of quantum mechanics.
Universe 2020, 6, 180; doi:10.3390/universe6100180 www.mdpi.com/journal/universe Universe 2020, 6, 180 2 of 25
One of the promising candidates for a theory of physical vacuum is superfluid vacuum theory (SVT), a post-relativistic approach to high-energy physics and gravity. Historically, it evolved from Dirac’s idea of viewing the physical vacuum as a nontrivial quantum object, whose phase and derived velocity are non-observable in a quantum-mechanical sense [4]. The term ‘post-relativistic’, in this context, means that SVT can generally be a non-relativistic theory; which nevertheless contains relativity as a special case, or limit, with respect to some dynamical value such as momentum (akin to general relativity being a superset of the Newton’s theory of gravity). Therefore, underlying three-dimensional space would not be physically observable until an observer goes beyond the above-mentioned limit, as will be discussed in more detail later in this article. The dynamics and structure of superfluid vacuum are being studied, using various approaches which agree upon the main paradigm (physical vacuum being a background quantum liquid of a certain kind, and elementary particles being excitations thereof), but differ in their physical details, such as an underlying model of the liquid [5–7]. It is important to work with a precise definition of superfluid, to ensure that we avoid the most common misconceptions which otherwise might arise when one attempts to apply superfluid models to astrophysics and cosmology, some details can be found in AppendixA. In fact, some superfluid-like models of dark matter based on classical perfect fluids, scalar field theories or scalar-tensor gravities, turned out to be vulnerable to experimental verification [8]. Moreover, superfluids are often confused not only with perfect fluids, but also with the concomitant phenomenon of Bose-Einstein condensates (BEC), which is another kind of quantum matter occurring in low-temperature condensed matter [9]. However, even though BEC’s do share certain features with superfluids, this does not imply that they are superfluidic in general. In particular, quantum excitations in laboratory superfluids that we know of have dispersion relations of a distinctive shape called the Landau “roton” spectrum. Such a shape of the spectral curve is crucial, as it ensures the suppression of dissipative fluctuations at a quantum level [10,11], which results in inviscid flow [12,13]. If plotted as an excitation energy versus momentum, the curve starts from the origin, climbs up to a local maximum (called the maxon peak), then slightly descends to a local nontrivial minimum (called the “roton” energy gap); then grows again, this time all the way up, to the boundary of the theory’s applicability range. In fact it is not the roton energy gap alone, but the energy barrier formed by the maxon peak and roton minimum in momentum space, which ensures the above-mentioned suppression of quantum fluctuations in quantum liquid and, ultimately, causes its flow to become inviscid. In other words, it is the global characteristics of the dispersion curve, not just the existence of a nontrivial local minimum and related energy gap, which is important for superfluidity to occur. Obviously, these are non-trivial properties, which cannot possibly occur in all quantum liquids and condensates. Further details and aspects are discussed in AppendixA. This paper is organized as follows. Theory of physical vacuum based on the logarithmic superfluid model is outlined in Section2, where we also demonstrate how four-dimensional spacetime can emerge from the three-dimensional dynamics of quantum liquid. In Section3, we derive the gravitational potential, induced by the logarithmic superfluid vacuum in a given state, using certain simplifying assumptions. Thereafter, in Section4, we give a brief physical interpretation of different parts of the derived gravitational potential and estimate their characteristic length scales. In Section5, profiles of induced matter density are derived and discussed for the case of spherical symmetry. Galactic scale phenomena are discussed in Section6, where the phenomenon of galactic rotation curves is explained without introducing any exotic matter ad hoc. In Section7, we discuss the various mechanisms of the accelerating expansion of the Universe, as well as the cosmological singularity, “vacuum catastrophe” and cosmological coincidence problems. Conclusions are drawn in Section8. Universe 2020, 6, 180 3 of 25
2. Logarithmic Superfluid Vacuum Superfluid vacuum theory assumes that the physical vacuum is described, when disregarding quantum fluctuations, by the fluid condensate wavefunction Ψ(r, t), which is a three-dimensional Euclidean scalar. The state itself is described by a ray in the corresponding Hilbert space, therefore this wavefunction obeys a normalization condition Z hΨ|Ψi = ρ dV = M, (1) V where M and V are the total mass and volume of the fluid, respectively, and ρ = |Ψ|2 is the fluid mass density. The wavefunction’s dynamics is governed by an equation of a U(1)-symmetric Schrödinger form: " # h¯ 2 −ih¯ ∂ − ∇2 + V (r, t) + F(|Ψ|2) Ψ = 0, (2) t 2m ext where m is the constituent particles’ mass, Vext(r, t) is an external or trapping potential and F(ρ) is a duly chosen function, which effectively takes into account many-body effects inside the fluid. This wave equation can be formally derived as a minimizing condition of an action functional with the following Lagrangian:
ih¯ h¯ 2 L = (Ψ∂ Ψ∗ − Ψ∗∂ Ψ) + |∇Ψ|2 + V (r, t) |Ψ|2 + V(|Ψ|2), (3) 2 t t 2m ext where V(ρ) equals to a primitive of F(ρ) up to an additive constant: F(ρ) = V0(ρ); throughout the paper the prime denotes a derivative with respect to the function’s argument. In this picture, massless excitations, such as photons, are analogous to acoustic waves propagating p 0 with velocity cs ∝ |p (ρ)|, where fluid pressure p = p(ρ) is determined via the equation of state. For the system (2), both the equation of state and speed of sound can be derived using the fluid-Schrödinger analogy, which was established for a special case in [14], and generalized for an arbitrary F(ρ) in works [7,15]. In a leading-order approximation with respect to the Planck constant, we obtain 1 Z 1 p = − ρF0(ρ) dρ, c2 = ρ|F0(ρ)|, (4) m s m while higher-order corrections would induce Korteweg-type effects, thus significantly complicating the subject matter [15]. Furthermore, it is natural to require that superfluid vacuum theory must recover Einstein’s theory of relativity at a certain limit. One can show that at a limit of low momenta of quantum excitations, often called a “phononic” limit by analogy with laboratory quantum liquids, Lorentz symmetry does emerge. This can be easily shown by virtue of the fluid/gravity analogy [16], which was subsequently used to formulate the BEC-spacetime correspondence [7]; it can also be demonstrated by using dispersion relations [11,17], which are generally become deformed in theories with non-exact Lorentz symmetry [18–21]. This correspondence states that Lorentz symmetry is approximate, while four-dimensional spacetime is an induced phenomenon, determined by the dynamics of quantum Bose liquid moving in Euclidean three-dimensional space. The latter is only observable by a certain kind of observer, a F(ull)-observer. Other observers, R(elativistic)-observers, perceive this superfluid as a non-removable background, which can be modeled as a four-dimensional pseudo-Riemannian manifold. What is the difference between these types of observers? F-observers can perform measurements using objects of arbitrary momenta and “see” the fundamental superfluid wavefunction’s evolution in three-dimensional Euclidean space according to Equation (2) or an analogue thereof. On the other hand, R-observers are restricted to measuring only small-momentum small-amplitude excitations of the background superfluid. This is somewhat Universe 2020, 6, 180 4 of 25 analogous to listening to acoustic waves (phonons) in the conventional Bose-Einstein condensates, but being unaware of higher-energy particles such as photons or neutrons. According to BEC-spacetime correspondence, a R-observer “sees” himself located inside four-dimensional curved spacetime with a pseudo-Riemannian metric. The latter can be written in Cartesian coordinates as [7]:
. 2 2 2 . − cs − η (∇S) . −η∇ S ρ gµν ∝ ·········· , (5) cs . −η∇S . I where η = h¯ /m, S = S(r, t) = −i ln (Ψ(r, t)/|Ψ(r, t)|) is a phase of the condensate wavefunction √ written in the Madelung representation, Ψ = ρ exp (iS), and I is a three-dimensional unit matrix. To maintain the correct metric signature in Equation (5), condition |cs| > η |∇S| must be imposed, which indicates that cs is the maximum attainable velocity of test particles (i.e., small-amplitude excitations of the condensate), moving along geodesics on this induced spacetime. Therefore, cs is the velocity of those excitations of vacuum, which describe massless particles in the low-momentum limit, whereas massive test particles move along geodesics of a pseudo-Riemannian manifold with metric Equation (5). According to a R-observer, they are freely falling, independently of their properties including their rest mass. In this approach, we interpret Einstein field equations not as differential equations for an unknown metric; but as a definition for an induced stress-energy tensor, describing some effective matter to which test particles couple. Therefore, this would be the gravitating matter observed by a R-observer. We thus obtain −1 1 Teµν ≡ κ Rµν(g) − gµνR(g) , (6) 2 = 2 where κ 8πG/c(0) is the Einstein’s gravitational constant. An example of usage of this procedure will be considered in Section 7.1. While Equation (6) is in fact an assumption, it should hold not only under the validity of conventional general relativity, but also in other Lorentz-symmetric theories of gravity which are linear with respect to the Riemann tensor, because the form of Einstein equations is quite universal (up to a conformal transform). For other Lorentz-symmetric theories, whose field equations cannot be transformed into this form, definition (6) can be adjusted accordingly. Furthermore, one can see from Equation (4), that cs contains an unknown function F(ρ). To determine its form, let us recall that one of the relativistic postulates implies that velocity cs should not depend on density, at least in the classical limit. More specifically, at low momenta, 10 −1 this velocity should tend to the value c(0) ≈ c, where c = 2.9979 × 10 cm s is called the speed of light in vacuum, for historical reasons. Recalling Equation (4), this requirement can be written as a differential equation [7]: 0 2 ρ|F (ρ)| = mcs ≈ const(ρ), (7) where const(ρ) denotes a function which does not depend on density. The solution of this differential equation is a logarithmic function: F(ρ) = −b ln (ρ/ρ¯), (8) where b and ρ¯ are generally real-valued functions of coordinates. The wave Equation (2) thus narrows down to " # h¯ 2 ih¯ ∂ Ψ = − ∇2 + V (r, t) − b ln (|Ψ|2/ρ¯) Ψ, (9) t 2m ext where b is the nonlinear coupling; b = b(r, t) in general. Correspondingly, Equation (4) yields q p = −(b/m)ρ, cs = |b|/m, (10) Universe 2020, 6, 180 5 of 25 thus indicating that logarithmic Bose liquid behaves like barotropic perfect fluid; but only when one neglects quantum corrections, and assumes classical averaging. This reaffirms the statement made in the previous Section about the place of perfect-fluid models when it comes to gravitational phenomena. The way gravity emerges in the superfluid vacuum picture is entirely different from those models, as will be demonstrated shortly, after we have specified our working model. Some special cases of Equation (9), for example when b → b0 = const, were extensively studied in the past, although not for reasons related to quantum liquids [22,23]. There were also extensive mathematical studies of these equations, to mention just some very recent literature [24–37]. Interestingly, wave equations with logarithmic nonlinearity can be also introduced into fundamental physics independently of relativistic arguments [7,38,39]. This nonlinearity readily occurs in the theory of open quantum systems, quantum entropy and information [40,41]; as well as in the theory of general condensate-like materials, for which characteristic kinetic energies are significantly smaller than interparticle potentials [42]. One example of such a material is helium II, the superfluid phase of helium-4. For the latter, the logarithmic superfluid model is known to have been well verified by experimental data [10,43]. Among other things, the logarithmic superfluid model does reproduce the sought-after Landau-type spectrum of excitations, discussed in the previous Section; detailed derivations can be found in [10]. One of underlying reasons for such phenomenological success is that the ground-state wavefunction of free (trapless) logarithmic liquid is not a de Broglie plane wave, but a spatial Gaussian modulated by a de Broglie plane wave. This explains the liquid’s inhomogenization followed by the formation of fluid elements or parcels; which indicates that such models do describe fluids, rather than gaseous matter [44–49]. To summarize, a large number of arguments to date, both theoretical and experimental, demonstrate the robustness of logarithmic models in the general theory of superfluidity. In the next Section we shall demonstrate the logarithmic superfluid model’s capabilities when assuming superfluidity of the physical vacuum itself. In what follows, we shall make use of a minimal inhomogeneous model for the logarithmic superfluid which was proposed in [42], based on statistical and thermodynamics arguments. In the F-observer’s picture, its wave equation can be written as " # h¯ 2 q |Ψ|2 ih¯ ∂ Ψ = − ∇2 + V (r, t) − b − ln Ψ, (11) t 2m ext 0 r2 ρ¯ √ where r = |r| = r · r is a radius-vector’s absolute value, and b0 and q are real-valued constants. For definiteness, let us assume that b0 > 0, because one can always change the overall signs of the nonlinear term F(ρ) and the corresponding field-theoretical potential V(ρ). As always, this wave equation must be supplemented with a normalization condition (1), boundary and initial conditions of a quantum-mechanical type; which ensure the fluid interpretation of Ψ [50]. 2 One can show that nonlinear coupling b = b(r) = b0 − q/r is a linear function of the quantum temperature TΨ, which is defined as a thermodynamic conjugate of quantum information entropy sometimes dubbed as the Everett-Hirschman information entropy. The latter can be written as 2 R 2 2 SΨ = −hΨ| ln (|Ψ| /ρ¯)|Ψi = − V |Ψ| ln (|Ψ| /ρ¯) dV, where a factor 1/ρ¯ is introduced for the sake of correct dimensionality, and can be absorbed into an additive constant due to the normalization condition (1). Therefore, one can expect that the thermodynamical parameters
b0 = b0(TΨ), q = q(TΨ) (12) are constant at a fixed temperature TΨ. Thus, for a trapless version of the model (11) we have four parameters, but only two of them, m and ρ¯, are a priori fixed, whereas the other two, b0 and q, can vary depending on the environment. Universe 2020, 6, 180 6 of 25
3. Induced Gravitational Potential Invoking model (11), while neglecting quantum fluctuations, let us assume that physical vacuum is a collective quantum state described by wavefunction Ψ = Ψvac(r, t), which forms a self-gravitating configuration with a center at r = 0. Therefore, for this state, the solution of Equation (11) is equivalent to the solution of the linear Schrödinger equation, " # h¯ 2 ih¯ ∂ Ψ = − ∇2 + V (r, t) Ψ, (13) t 2m eff for a particle of mass m driven by an effective potential
|Ψ (r, t)|2 V (r, t) = V (r, t) − b ln vac eff ext ρ¯ (14) q |Ψ (r, t)|2 = V (r, t) − b − ln vac , ext 0 r2 ρ¯ when written in Cartesian coordinates [42]. If working in curvilinear coordinates, the last formula must be supplemented with terms which arise after separating out the angular variables in the wave equation. In the absence of quantum excitations and other interactions, it is natural to associate this effective quantum-mechanical potential with the only non-removable fundamental interaction that we know of: gravity. This interpretation will be further justified in Section4. Therefore, in Cartesian coordinates one can write the induced gravitational potential as
1 1 q |Ψ (r, t)|2 Φ(r, t) = − V (r, t) = b − ln vac , (15) m eff m 0 r2 ρ¯ where we assume that the background superfluid is trapless, i.e., we set Vext = 0. It should also be remembered that in curvilinear coordinates, this formula must be modified according to the remark after Equation (14); but for now we shall disregard any anisotropy and rotation. It should be noticed that if one regards this potential as a multiplication operator then its quantum-mechanical average would be related to the Everett-Hirschman information entropy 2 discussed in the previous Section: hΦi ∼ TΨ hΨ| ln (|Ψ| ) |Ψi ∼ TΨSΨ. This not only makes theories of entropic gravity (which are essentially based on the ideas of Bekenstein, Hawking, Jacobson and others) a subset of the logarithmic superfluid vacuum approach, but also endows them with an underlying physical meaning and origin of the entropy implied. We can see that the induced potential maintains its form as long as the physical vacuum stays in the state |Ψvaci. If the vacuum were to transition into a different state, then it would change its wavefunction; hence the induced gravitational potential would also change. We expect that our vacuum is currently in a stable state, which is close to a ground state or at least to a metastable state, with a sufficiently large lifetime. It is thus natural to assume that the state |Ψvaci is stationary and rotationally invariant. As we established earlier, the wavefunction describing such a state should be the solution of a quantum wave equation containing logarithmic nonlinearity. In the case of trivial spatial topology and infinite extent, the amplitude of such a solution is known to be the product of a Gaussian function, which was mentioned in the previous Section, and a conventional quantum-mechanical part, which is a product of an exponential function, power function and a polynomial. Thus we can write the amplitude’s general form as:
χ /2 p r 0 a2 2 a1 a0 |Ψvac| = ρ¯ P(r) exp − r + r + , (16) `¯ 2`¯2 2`¯ 2 Universe 2020, 6, 180 7 of 25
1/3 where P(r) is a polynomial function, χ0 and a’s are constants, and `¯ = (m/ρ¯) is a classical `¯ characteristic length scale of the logarithmic√ nonlinearity (alternatively, one can choose being equal to the quantum characteristic length, h¯ / mb0, which might be more useful for h¯ -expansion techniques). If quantum liquid occupies an infinite spatial domain then the normalization condition (1) requires
a2 > 0, (17) which is also confirmed by analytical and numerical studies of differential equations with logarithmic nonlinearity of various types [23–25,28,31,35,42]. Both the form of a function P(r) and the values of χ0 and a’s must be determined by a solution of an eigenvalue problem for the wave equation under normalization and boundary conditions. At this stage, those conditions are not yet precisely known; even if they were, we do not yet know which quantum state our vacuum is currently in. Therefore, these constants’ values remain theoretically unknown at this stage, yet can be determined empirically. Furthermore, for the sake of simplicity, let us approximate the power-polynomial term (r/`¯)χ0/2P(r), by the single power function (r/`¯)χ/2, where the constant χ is the best fitting parameter. Therefore, we can approximately rewrite Equation (16) as h a a r i |Ψ |2 ≈ ρ¯ exp − 2 r2 + 1 r + χ ln + a , (18) vac `¯2 `¯ `¯ 0 which is more convenient for further analytical studies than the original expression (16). From the empirical point of view, the function (18) can be considered as a trial function, whose parameters can be fixed using experimental data following the procedure we describe below. For the trial solution (18), the normalization condition (1) immediately imposes a constraint for one of its parameters:
(χ+3)/2 −1 Ma2 3 1 a1 3 exp(a0) ≈ Z˘ , + √ Z˘ 2, , (19) 2πm 2 2 a2 2
˘ 2 where we introduced an auxiliary function Z (a, b) = Γ (a + χ/2) 1F1 a + χ/2, b; a1/4a2 , where Γ(a) and 1F1(a, b; z) are the gamma function and Kummer confluent hypergeometric function, respectively. If values of a’s and χ are determined, e.g., empirically, then this formula can be used to estimate the ratio M/m. Furthermore, by substituting the trial solution (18) into the definition (15), we derive the induced gravitational potential as a sum of seven terms:
Φ(r) = Φsmi(r) + ΦRN(r) + ΦN(r) (20) +Φgal(r) + Φmgl(r) + ΦdS(r) + Φ0, Universe 2020, 6, 180 8 of 25 where
χ q ln (r/`¯) L2 ln (r/`¯) Φ (r) = − = −ζ c2 smi , (21) smi m r2 χq b r2 a q 1 L2 Φ (r) = − 0 = −ζ c2 RN , (22) RN m r2 a0q b r2 a q 1 GM Φ (r) = − 1 = − , (23) N m`¯ r r χ b Φ (r) = 0 ln (r/`¯) = c2 χ ln (r/`¯), (24) gal m b a1b0 2 r Φmgl(r) = r = ζa1 cb , (25) m`¯ Lmgl a b r2 Φ (r) = − 2 0 r2 = −c2 , (26) dS `¯2 b 2 m LdS and ! 1 a q 1 q Φ = a b + 2 = a b + (27) 0 m 0 0 `¯2 m 0 0 2 LdS is the additive constant. Here, and throughout the paper, we denote the sign functions by ζ’s: ζα = sign (α), and use the following notations:
r s b0 a1q |χ q| cb = , GM = , Lsmi = , m m`¯ b0 s |a0q| `¯ |q| LRN = , Lmgl = = , (28) b0 |a1| mGM `¯ χ`¯ χq LdS = √ , Lχ = = , a2 a1 mGM where G is the Newton’s gravitational constant as per usual. Furthermore, Lorentz symmetry emerges in the “phononic” low-momentum limit of the theory, as discussed in the previous Section. Therefore, a R-observer would perceive the gravity induced by potential (20) as curved four-dimensional spacetime, which is a local perturbation (not necessarily small) of the background flow metric, such as the one derived in Section 5.3 of [7], see Section 7.1 below. In a rotationally invariant case, the line element of this spacetime can be written in the Newtonian ( ) 2 gauge; if Φ r /c(0) 1, then it can be approximately rewritten in the form " # 2Φ(r) dr2 ds2 ≈ −c2 1 + dt2 + + R2(r)dσ2, (29) (0) 2 + ( ) 2 c(0) 1 2Φ r /c(0) h i ( ) = + O ( ) 2 ≈ 2 = 2 + 2 2 where R r r 1 Φ r /c(0) r, d σ dθ sin θ dϕ is the line element of a unit two-sphere, and a leading-order approximation with respect to the Planck constant is implied, as usual. The mapping (29) is valid for regions where the induced metric maintains a signature ‘− + ++’, and its matrix is non-singular. In other regions, such as close vicinities of spacetime singularities or horizons, the relativistic approximation is likely to fall outside its applicability range, thus it should be replaced with the F-observer’s description of reality. The main simplifying assumptions and approximations underlying the derivation of our gravitational potential are summarized and enumerated in the AppendixB. Universe 2020, 6, 180 9 of 25
4. Physical Interpretation It should be noticed that if we did not have a logarithm in the original model (11), then in Equations (14) and (15), then we would not have arrived at the polynomial functions in Equations (20)–(26), which are easily recognizable. This reaffirms our expectations that the underlying model can be successfully confirmed by experiment; but first those functions must be endowed with precise physical meaning. In this Section, we shall assign a physical interpretation to each term of the derived gravitational potential. For the sake of brevity, we shall be omitting an additive constant Φ0, assuming that it is 2 small compared to c(0).
4.1. Potential ΦN and Gravitational Mass Generation We begin with term (23), which has the most obvious meaning. In a non-relativistic picture, it represents Newton’s model of gravity. According to the BEC-spacetime correspondence manifested through the mapping (29), a R-observer can observe an effect of the potential ΦN by measuring probe particles moving along geodesics in the Schwarzschild spacetime:
2 2 2 rH 2 dr 2 2 ds(N) ≈ −c(0) 1 − dt + + r dσ , (30) r 1 − rH/r
= 2 where rH 2GM/c(0) is the Schwarzschild radius. Therefore, in absence of asymptotically non-vanishing terms, M can be interpreted as the gravitational mass of the configuration. This mass can be expressed in terms of superfluid parameters as ( 2a q 1 gravity, r = 1 , sign (a q) = (31) H 2 `¯ 1 −1 anti-gravity, mc(0) thus assigning physical meaning to a combination of parameters a1q/m`¯. In particular, one can see that a sign of the product a1q determines whether the ΦN interaction is attractive (gravity) or repulsive (anti-gravity). For most systems that we know of, anti-gravitational effects have not yet been observed, therefore one can assume that M > 0 or a1q > 0 (32) from now on. Nevertheless, it should be emphasized that the anti-gravity case is not a priori forbidden in superfluid vacuum theory. Indeed, the spacetime singularity occurs at r = 0 in a relativistic picture only, which poses certain issues for a R-observer, especially in the case of anti-gravity when a singularity is not covered by an event horizon (the existence of naked singularities is often doubted, on grounds of the cosmic censorship hypothesis). However, a F-observer would see no singular behavior in either case, because the wavefunction Ψvac remains regular and normalizable at each point of space and at any given time—as it should be in a quantum-mechanical theory. This reaffirms the fact that spacetime singularities are an artifact of incomplete information accessible to observers operating with relativistic particles [7]. Thus, the mapping from Equations (23)–(30) can be used to reformulate black hole phenomena in the language of continuum mechanics and the theory of superfluidity; which can resolve certain long-standing problems occurring in the relativistic theory of gravity. For instance, neglecting asymptotically non-flat terms for simplicity, one can view Equations (23), (30) and (31) as the gravitational mass generation mechanism: such mass is not a fundamental notion, but a composite quantum phenomenon induced by the background superfluid’s dynamics (through the elementary inertial mass m and critical density ρ¯), its quantum temperature (through q), and an exponential part of the condensate’s wavefunction (through a1). Such a mechanism can be thus considered as the Universe 2020, 6, 180 10 of 25
quantum-mechanical version of the Mach principle [7]. For example, if either a1 or q vanish, then the system would not possess any gravitational mass, but it still would be gravitating in a non-Newtonian way, if other potentials from Equation (20) are non-zero.
4.2. Potential ΦRN and Abelian Charges Equation (22) represents another potential which can be easily recognized. According to the mapping (29), potential ΦRN is observed by a R-observer as Reissner–Nordström spacetime, when taken together with the ΦN potential:
2 ! 2 r rQ dr ds2 ≈ −c2 1 − H + dt2 + + r2dσ2, (33) (N+RN) (0) 2 2 2 r r 1 − rH/r + rQ/r where rQ is a characteristic length scale: v √ c u 2|a q| r = 2L b = u 0 , (34) Q RN c t 2 (0) mc(0) provided a0q < 0. The Reissner–Nordström metric is known to be associated with the gravitational field caused by a charge related to an abelian group. An example would be an electric charge Q, which is related to the abelian group U(1) of electromagnetism. This charge can be thus revealed through the formula
2 4 2 rQc(0) 2|a q|c Q2 = ≈ 0 , (35) keG keGm where ke is the Coulomb constant. In other words, potential ΦRN describes, together with ΦN, the gravitational field created by an object of a charge Q and gravitational mass M. Thus, from a F-observer’s viewpoint, an electrical charge is not an elementary notion, but a composite quantum phenomenon, induced by the background superfluid’s dynamics (through the elementary inertial mass m), its quantum temperature (through q), and overall constant coefficient of the condensate’s wavefunction (through a0). It should be noted also that ΦRN is a short-range potential, therefore, it becomes substantial only 2 ¯ at those microscopical length scales, of an order rQH = rQ/rH = ` |a0/a1| or below. Since rQ < rH for most objects we know of, we have 0 6 rQH < rH. Thus, those scales would be causally inaccessible to a R-observer, but a F-observer would have no problem accessing them, per usual.
4.3. Potential Φsmi and Strong Gravity As the distance from a gravitating center decreases, it is term (21) which eventually predominates. According to the mapping (29), this potential Φsmi, when taken together with the ΦN and ΦRN potentials, induces spacetime with the line element: r2 r2 2 ≈ − 2 − rH + Q − W r 2 ds(N+RN+smi) c(0) 1 r r2 ζχq r2 ln `¯ dt dr2 (36) + + r2dσ2, 2 2 2 ¯ 2 1 − rH/r + rQ/r − ζχqrW ln r/` /r where rW is a characteristic length scale: v √ c u 2|χ q| r = 2L b = u . (37) W smi c t 2 (0) mc(0) Universe 2020, 6, 180 11 of 25
The potential Φsmi has a distinctive property: unlike other sub-Newtonian potentials in Equation (20), it can switch between repulsive and attractive regimes, depending on whether the distance is larger or smaller than `¯. The magnitudes of Φsmi and ΦRN become comparable at two values of r, shown by the formula
(±) ¯ 2 2 ¯ rWQ = ` exp (±rQ/rW ) = ` exp (±|a0/χ|), (38)
(−) (+) which indicates that |Φsmi| overtakes |ΦRN| either at r < rWQ or at r > rWQ. If |a0/χ| is large then (+) (−) rWQ is exponentially large and rWQ is exponentially small. Magnitudes of Φsmi and ΦN become comparable at a certain value of r, which is shown by the formula